Theorem sb9 1896

Description: Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.)

Assertion

Ref	Expression
sb9	|- ( A. x [ x / y ] ph <-> A. y [ y / x ] ph )

Proof of Theorem sb9

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sb9v 1895	. . 3 |- ( A. y [ y / w ] [ w / x ] ph <-> A. w [ w / y ] [ w / x ] ph )
2		sbcom 1890	. . . 4 |- ( [ w / y ] [ w / x ] ph <-> [ w / x ] [ w / y ] ph )
3	2	albii 1399	. . 3 |- ( A. w [ w / y ] [ w / x ] ph <-> A. w [ w / x ] [ w / y ] ph )
4		sb9v 1895	. . 3 |- ( A. w [ w / x ] [ w / y ] ph <-> A. x [ x / w ] [ w / y ] ph )
5	1, 3, 4	3bitri 204	. 2 |- ( A. y [ y / w ] [ w / x ] ph <-> A. x [ x / w ] [ w / y ] ph )
6		ax-17 1459	. . . 4 |- ( ph -> A. w ph )
7	6	sbco2h 1879	. . 3 |- ( [ y / w ] [ w / x ] ph <-> [ y / x ] ph )
8	7	albii 1399	. 2 |- ( A. y [ y / w ] [ w / x ] ph <-> A. y [ y / x ] ph )
9	6	sbco2h 1879	. . 3 |- ( [ x / w ] [ w / y ] ph <-> [ x / y ] ph )
10	9	albii 1399	. 2 |- ( A. x [ x / w ] [ w / y ] ph <-> A. x [ x / y ] ph )
11	5, 8, 10	3bitr3ri 209	1 |- ( A. x [ x / y ] ph <-> A. y [ y / x ] ph )

Syntax hints:   <-> wb 103   A.wal 1282   [wsb 1685

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686

This theorem is referenced by:  sb9i  1897